MobiWave: Dispatch-Oriented Graph Wavelets and Drift-Guided Selective Optimization for Autonomous Fleet Rebalancing Xiao Han, Pinbo Wang, Yuanshao Zhu, Guojiang Shen, Xiangjie Kong Autonomous fleets enable mobility platforms to coordinate idle vehicles directly, making fleet-wide rebalancing possible. However, two obstacles limit reliable deployment: overlapping regional and local traffic patterns can hide roads that remain useful for dispatch, and mobility drift can make a trained policy unreliable. Existing spatial aggregation mixes these patterns, while updating all parameters from limited recent data is costly and can damage stable knowledge. We propose MobiWave, a framework that connects a dispatchoriented multi-scale graph wavelet module with Drift-Guided LayerSelective Optimization (DGLS). The first module addresses the representation challenge by separating graph-frequency patterns and weighting each scale according to its value for demand prediction and feasible rebalancing. DGLS addresses the adaptation challenge by measuring Dispatch-weighted Spectral Drift, selecting affected layers within a resource budget, and separating short shocks from persistent changes through a drift-aware fast–slow update. Candidate validation further rejects updates that fail to improve held-out dispatch reward without worsening monitored service or safety constraints. Experiments on both real-world datasets and simluated environments demonstrate the effectiveness of MobiWave in comparing with state-of-the-art methods. The source code and datasets are available at https://anonymous.4open.science/r/MobiWave-40F8/.
CCS Concepts • Information systems → Spatial-temporal systems; • Applied computing → Transportation.
Keywords Autonomous fleet rebalancing, graph wavelets, urban mobility drift, continual adaptation, selective optimization ACM Reference Format: Xiao Han, Pinbo Wang, Yuanshao Zhu, Guojiang Shen, Xiangjie Kong. 2027. MobiWave: Dispatch-Oriented Graph Wavelets and Drift-Guided Selective Optimization for Autonomous Fleet Rebalancing. In Proceedings of the 33rd ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD ’27). ACM, New York, NY, USA, 12 pages.
1
Introduction
Urban mobility platforms commonly assign incoming requests to nearby drivers [4, 12, 21]. Because drivers may favor familiar areas or particular trips, the resulting dispatch is not fully controlled by the platform [4, 19]. The platform must therefore model networkwide supply–demand conditions as well as drivers’ responses to dispatch instructions. Autonomous vehicles change this setting because they can carry out feasible platform decisions directly, making driver behavior no longer the main source of uncertainty [24, 35]. Dispatch can then shift from driver-constrained order matching to KDD ’27, San Jose, CA, USA 2027.
Fine-grained state is important !
Adaptive layer updates are needed ! Road Reconfiguration
Layer 1 Layer 2
Free-flowing Side Roads
…
arXiv:2607.24365v1 [cs.LG] 27 Jul 2026
Abstract
Layer n Congested Area
(a) Fine-Grained Road States
FunctionalZone Shift
LLM
Fixed Updating
(b) Adaptive Layer Updates
Figure 1: Motivation for MobiWave: (a) Broad congestion can hide free-flowing side roads that are useful for rebalancing. (b) Road reconfiguration and functional-zone shifts can affect different model layers, while fixed updates may miss them.
fleet-wide rebalancing that maximizes long-term system value. This shift makes two abilities critical: extracting road-network information that supports rebalancing [18, 28] and maintaining reliable dispatch decisions [31] as urban mobility conditions evolve. Both abilities are difficult because urban mobility changes at several scales. Regional supply and demand follow hourly, daily, and weekly cycles, and their changes can spread across connected regions [12]. Accidents, events, and temporary construction can instead produce sharp local changes [36]. These patterns overlap in space and time, making it difficult to identify their distinct effects on fleet decisions. For example, a city-wide rush hour and a short road closure can both create nearby vehicle shortages, although they call for different rebalancing actions. Over longer periods, new roads and urban rezoning reshape connectivity, while changes in residents’ travel habits shift regional demand [17]. These changes weaken models trained on historical data. Retraining on the complete history is costly, whereas updating every parameter from a short recent window can be unsafe. We therefore ask how an autonomous fleet can extract dispatch-critical road-network information and adapt its policy from limited recent data. The first part of this question concerns the road-network representation provided to the dispatcher. Existing end-to-end spatiotemporal models commonly mix road-network signals through repeated aggregation over neighboring regions [1, 20, 33]. Figure 1(a) illustrates how this aggregation can blur local conditions. Although congestion appears to spread around the accident site, many nearby side streets remain free-flowing. Treating the entire area as congested would hide viable routes and lead to poor vehicle rebalancing. Graph spectral methods offer a way to separate road-network signals by frequency [8, 11, 22, 30], and recent two-dimensional filters model joint spatial and temporal spectral relations [6]. Yet these methods do not identify which graph-frequency components are most useful for dispatch in each region and time step. Therefore,
KDD ’27, August 2027, San Jose, CA, USA
the first challenge is to separate overlapping road-network patterns across scales and retain the components that support vehicle rebalancing. Accurately representing dispatch-relevant road-network conditions and maintaining a reliable dispatch model under evolving mobility conditions are equally important. Large spatiotemporal dispatch models learn stable mobility relations from extensive historical data. Some of these relations become outdated when roads are reconfigured, urban functions shift, or residents change where and when they travel. A recent window better reflects current conditions, yet it is usually too small to support a safe update of every parameter. Periodic full-model tuning is also costly in time and memory and can overfit a short abnormal window. Distribution tests can reveal changes between recent and historical states [10], while regularization and replay can preserve earlier knowledge [15, 26]. Recent studies reduce adaptation cost by updating selected parameters, routing data through adaptive experts, or maintaining optimizer states at several time scales [2, 14, 23, 27, 37]. Figure 1(b) illustrates that road reconfiguration and functional-zone shifts may affect different internal representations, which a fixed update rule can miss. A reliable update must therefore connect each observed road-network change to the model components that control the affected dispatch decisions. Existing approaches do not jointly measure dispatch-relevant drift, choose affected layers within a resource budget, preserve stable knowledge, and reject harmful updates. Therefore, the second challenge is to locate and update only the model components affected by current mobility drift, using limited recent data without erasing historical knowledge. To address these challenges, we propose MobiWave, which combines dispatch-oriented graph wavelets with drift-guided selective optimization for autonomous fleet rebalancing. To address the first challenge, the dispatch-oriented multi-scale graph wavelet module combines causal temporal features with graph wavelets to separate city-wide trends, cross-region propagation, and local disruptions. Dispatch-aware gating then weights these graph-frequency scales according to their value for rebalancing. To address the second challenge, DGLS measures Dispatch-weighted Spectral Drift, selects affected layers within a resource budget, and uses drift-aware fast– slow updates and candidate validation to protect stable knowledge. The dispatch evidence learned by the first module thus supports both current rebalancing and selective adaptation, linking the two challenges within one framework. Our contributions are summarized as follows: • We formulate autonomous fleet rebalancing under urban mobility drift as a joint problem of dispatch-critical state perception and continual model adaptation. • We introduce a dispatch-oriented multi-scale graph wavelet module that separates road dynamics at multiple scales and learns scale weights for demand prediction and fleet control. • DGLS is developed to measure Dispatch-weighted Spectral Drift, select and adapt affected layers within a fixed budget, and reject updates that fail candidate validation. • Experiments on both real-world datasets and the simulation platform demonstrate the effectiveness of MobiWave in reducing empty-loaded rate and improving profit.
X. Han, P.B. Wang, Y.S. Zhu, G.J. Shen, and X.J. Kong
2
Preliminaries
Road-network representation. We divide a city into 𝑁 nonoverlapping dispatch zones and represent their one-step reachability at time 𝑡 by an undirected weighted graph G𝑡 = (V, E𝑡 , W𝑡 ), where V = {𝑣 1, . . . , 𝑣 𝑁 } is the zone set. The symmetric weights combine feasible reachability, free-flow travel time, and historical Í bidirectional origin–destination flow. With [D𝑡 ] 𝑖𝑖 = 𝑗 [W𝑡 ] 𝑖 𝑗 , the normalized Laplacian L𝑡 = I − D𝑡−1/2 W𝑡 D𝑡−1/2 defines graph spatial frequency; the inverse degree is set to zero for an isolated zone. A road closure or new connection updates E𝑡 , W𝑡 , and L𝑡 . Traffic and fleet states. At dispatch time 𝑡, X𝑡 ∈ R𝑁 ×𝐹 stacks the causal zone states x𝑖,𝑡 = [𝑑𝑖,𝑡 , 𝑛𝑖,𝑡 , 𝑏𝑖,𝑡 , 𝑠𝑖,𝑡 , 𝝉𝑖,𝑡 , 𝒄𝑖,𝑡 , e𝑖,𝑡 ], including current demand, available vehicles, request backlog, passenger assignments, candidate-move travel times and costs, and observed external factors. The state also contains in-transit records, which the proposed module summarizes as expected arrivals a𝑡tr . Every feature used at time 𝑡 is observed no later than 𝑡. Rebalancing action. The platform first assigns 𝑠𝑖,𝑡 vehicles to passenger orders and then controls the remaining idle vehicles 𝑛¯𝑖,𝑡 = 𝑛𝑖,𝑡 −𝑠𝑖,𝑡 through an integer flow matrix U𝑡 = [𝑢𝑖 𝑗,𝑡 ] ∈ N0𝑁 ×𝑁 . The entry 𝑢𝑖𝑖,𝑡 denotes vehicles that stay in zone 𝑖, while 𝑢𝑖 𝑗,𝑡 for 𝑖 ≠ 𝑗 denotes vehicles sent from 𝑖 to an adjacent zone 𝑗. Feasible actions must satisfy 𝑢𝑖 𝑗,𝑡 = 0 if 𝑖 ≠ 𝑗 and (𝑣𝑖 , 𝑣 𝑗 ) ∉ E𝑡 ,
𝑁 ∑︁
𝑢𝑖 𝑗,𝑡 = 𝑛¯𝑖,𝑡 .
(1)
𝑗=1
Fleet conservation additionally accounts for vehicles arriving from earlier moves and completed passenger trips; in-transit vehicles remain in the fleet state until each arrives. Problem definition. Let 𝑟𝑡 denote the one-step dispatch reward, defined as passenger fare revenue minus the costs of empty travel, passenger waiting, and canceled requests. Given policy 𝜋𝜃 , autonomous fleet rebalancing seeks max 𝐽 (𝜃 ) = E𝜋𝜃 𝜃
"𝑇 −1 ∑︁
# 𝑡
𝛾 𝑟𝑡
𝑠.𝑡 . (1) and fleet conservation, (2)
𝑡 =0
where 𝛾 ∈ (0, 1]. We study the harder case in which the data distribution changes over time and only a small recent window and a fixed update budget are available.
3 Methodology 3.1 Framework Overview Figure 2 presents MobiWave, which maps the road graph, causal road-network history, and current fleet distribution to feasible rebalancing decisions through two connected modules. The dispatchoriented multi-scale graph wavelet module separates broad and local graph-frequency patterns and fuses the components that support demand prediction and rebalancing. The resulting scale features and gate weights also provide dispatch evidence for DGLS, which detects relevant mobility drift, selects and adapts affected layers under a resource budget, and validates every candidate update before deployment. The shared representation therefore supports both current dispatch and controlled adaptation.
MobiWave: Dispatch-Oriented Graph Wavelets and Drift-Guided Selective Optimization for Autonomous Fleet Rebalancing
KDD ’27, August 2027, San Jose, CA, USA
Figure 2: The MobiWave framework. The dispatch-oriented multi-scale graph wavelet module supports fleet rebalancing. DGLS measures dispatch-weighted spectral drift, selectively updates affected layers under a resource budget, and applies candidate validation before deployment.
3.2
Dispatch-Oriented Multi-scale Graph Wavelet
Reliable rebalancing requires a road-network representation that separates broad traffic patterns from local road changes and identifies which pattern matters at each zone and time. Existing neighborhood aggregation can mix these signals and hide useful local conditions, whereas graph wavelets separate graph-frequency components while retaining their locations. Combining graph wavelets with causal temporal features and dispatch-aware gating allows the model to select the components that support fleet decisions. We therefore use a dispatch-oriented multi-scale graph wavelet module to construct and fuse scale features for fleet dispatch. Constructing this representation from a single state is insufficient because it cannot distinguish temporary from persistent imbalance, while future-dependent summaries would invalidate online dispatch. To expose only changes observed by time 𝑡, the module combines the current state, multi-horizon temporal summaries, periodic code p𝑡 , neighborhood demand–supply gap 𝜹𝑡nbr , and vehicles scheduled to arrive a𝑡tr : Z𝑡 = X𝑡 ∥ [M𝑡(ℎ) ∥𝚫𝑡(ℎ) ] 1𝑁 p𝑡⊤ 𝜹𝑡nbr a𝑡tr W𝑧 + 1𝑁 b𝑧⊤, (3) ℎ∈ H
where, for each horizon ℎ ∈ H , the current-window mean and its change from the immediately preceding window are M𝑡(ℎ) = Í2ℎ−1 (ℎ) 1 Íℎ−1 = M𝑡(ℎ) − ℎ1 𝑞=ℎ X𝑡 −𝑞 . The paired windows 𝑞=0 X𝑡 −𝑞 , 𝚫𝑡 ℎ describe both the current level and its recent direction at several
horizons. Because every time index is at most 𝑡, Equation (3) distinguishes short-lived changes from persistent imbalances without using future observations. Although Equation (3) uses only causal observations, directly aggregating Z𝑡 over neighboring zones could still mix broad traffic patterns with fine-grained road changes. We therefore decompose the input over multiple graph-frequency scales before dispatchspecific fusion. Let L𝑡 = Q𝑡 𝚲𝑡 Q𝑡⊤ be the normalized Laplacian of the current road graph and B = {1, . . . , 𝐾 } be the set of spectral scales. For each 𝑏 ∈ B, the localized graph-wavelet feature is H𝑏𝑡 = LN ReLU 𝑔𝑏 (L𝑡 )Z𝑡 W𝑏 + 1𝑁 b𝑏⊤ , 𝑏 ∈ B, (4) where 𝑔𝑏 (L𝑡 ) = Q𝑡 𝑔𝑏 (𝚲𝑡 )Q𝑡⊤ applies the kernel of scale 𝑏. Given ordered heat scales 𝜉 1 > · · · > 𝜉𝐾 −1 > 0, these kernels are exp(−𝜉 1 𝜆), 𝑏 = 1, 𝑔𝑏 (𝜆) = exp(−𝜉𝑏 𝜆) − exp(−𝜉𝑏 −1 𝜆), 2 ≤ 𝑏 ≤ 𝐾 − 1, 1 − exp(−𝜉 𝐾 −1 𝜆), 𝑏 = 𝐾.
(5)
The nonnegative kernels sum to one at every graph frequency, so the resulting features jointly cover the full spectrum. Low-frequency scales capture smooth regional patterns, whereas high-frequency scales preserve fine-grained local changes that conventional neighborhood aggregation may obscure. Direct eigendecomposition would be too costly whenever the road graph changes. With e L𝑡 = L𝑡 − I, all scales instead reuse a
KDD ’27, August 2027, San Jose, CA, USA
X. Han, P.B. Wang, Y.S. Zhu, G.J. Shen, and X.J. Kong
shared Chebyshev basis: 𝑔𝑏 (L𝑡 )Z𝑡 ≈
𝑃∑︁ cheb
𝑐𝑏,𝑝𝑇𝑝 (e L𝑡 )Z𝑡 , 𝑏 ∈ B,
(6)
from collapsing before the task losses reveal which scale mixture supports rebalancing. The learned scale features and gate weights also provide dispatch evidence for DGLS.
𝑝=0
3.3 where 𝑇𝑝 (·) is the 𝑝-th Chebyshev polynomial, with 𝑇0 (e L𝑡 )Z𝑡 = Z𝑡 , 𝑇1 (e L𝑡 )Z𝑡 = e L𝑡 Z𝑡 , and the standard recurrence for 𝑝 ≥ 2. The bandspecific coefficients 𝑐𝑏,𝑝 differ, while the sparse polynomial bases are computed once and reused across all bands. This preserves multi-scale filtering without per-step eigendecomposition. Separating the spectrum is not enough, because the useful scale varies across zones and time and a fixed average would mix the signals again. To relate each scale to the local fleet imbalance, a node-wise gate computes ∑︁ 𝑏 𝑏 h𝑖,𝑡 = Wres z𝑖,𝑡 + 𝛼𝑖,𝑡 H𝑖,𝑡 , 𝑏∈B
𝑚𝑏𝑡 exp 𝑓𝑔𝑏 ([H𝑏𝑖,𝑡 ∥𝜓𝑖,𝑡 ∥e𝑖,𝑡 ]) 𝑏 𝛼𝑖,𝑡 = ∑︁
𝑏′
𝑏′
𝑏′
𝑚𝑡 exp 𝑓𝑔 ([H𝑖,𝑡 ∥𝜓𝑖,𝑡 ∥e𝑖,𝑡 ])
(7) ,
𝑏′ ∈ B
where z𝑖,𝑡 is row 𝑖 of Z𝑡 , e𝑖,𝑡 contains observed external factors, and 𝜓𝑖,𝑡 = (𝑏𝑖,𝑡 + 𝑑𝑖,𝑡 − 𝑛¯𝑖,𝑡 )/(𝑛¯𝑖,𝑡 + 1) is the observed dispatch pressure. 𝑏 measures the dispatch relevance of scale 𝑏, while The weight 𝛼𝑖,𝑡 the residual path preserves information that should not be replaced by any single scale. The mask 𝑚𝑏𝑡 applies scale dropout in training while keeping one scale active; 𝑚𝑏𝑡 = 1 for all 𝑏 at inference. A dispatch representation is useful only if it can produce an executable action. A continuous action head can assign fractional vehicles or unreachable destinations, so MobiWave converts predicted shortage and move value into adjacency-masked probabilities and integer flows: (𝑢𝑖 𝑗,𝑡 ) 𝑗 ∈ N𝑖+ = Allocate 𝑛¯𝑖,𝑡 , (𝑝𝑖 𝑗,𝑡 ) 𝑗 ∈ N𝑖+ , (8) where 𝑝𝑖 𝑗,𝑡 = Softmax 𝑗 ∈ N𝑖+ 𝑓𝜋 (y𝑖𝜋𝑗,𝑡 ) , y𝑖𝜋𝑗,𝑡 is the per-vehicle move 𝑔 𝑗,𝑡 −b 𝑔𝑖,𝑡 ∥b 𝜌𝑖 𝑗,𝑡 ], 𝜌b𝑖 𝑗,𝑡 = 𝑓𝜌 ([h𝑖,𝑡 ∥h 𝑗,𝑡 ∥ ·b 𝑔 𝑗,𝑡 − return: y𝑖𝜋𝑗,𝑡 = [h𝑖,𝑡 ∥h 𝑗,𝑡 ∥b ⊤b 𝑔b𝑖,𝑡 ∥𝜏𝑖 𝑗,𝑡 ∥𝑐𝑖 𝑗,𝑡 ]), 𝑔b𝑖,𝑡 is the predicted fleet gap 𝑔b𝑖,𝑡 = 𝑏𝑖,𝑡 +𝑑𝑖,𝑡 +1𝐻 d𝑖,𝑡 − b b 𝜇𝑖 𝑛¯𝑖,𝑡 , and d𝑖,𝑡 is the demand forecast d𝑖,𝑡 = softplus(W𝑑 h𝑖,𝑡 + b𝑑 ). N𝑖+ contains zone 𝑖 and its reachable neighbors, and 𝜇𝑖 is the expected demand served by one available vehicle over the 𝐻 -step horizon. The masked softmax assigns zero probability to unreachable destinations, while Allocate rounds 𝑛¯𝑖,𝑡 𝑝𝑖 𝑗,𝑡 by largest remainders. Consequently, Equation (8) produces nonnegative, integer, adjacency-valid, and fleet-conserving actions. Training the prediction and policy heads separately could favor scales that fit demand but do not improve rebalancing. We therefore train the representation and dispatch policy jointly: Ljoint = 𝜆𝑑 Ldemand + 𝜆𝜌 Lreturn + 𝜆𝜋 Lpolicy 2 ∑︁ 1 𝑏 + 𝜆bal 𝛼 − , 𝐾
(9)
𝑏∈B
𝑏
where 𝛼 is the gate weight averaged over training zones and times. The demand and return losses provide direct supervision, while the clipped policy loss aligns the representation to dispatch reward. Together with scale dropout, the weak balance term prevents the gate
Drift-Guided Layer-Selective Optimization (DGLS)
The preceding representation supports current rebalancing, yet a policy trained on old data can become unreliable when mobility patterns change. Full-model updates may overwrite stable knowledge, while fixed parameter subsets may miss the layers affected by the current drift. The scale features expose changes that matter to dispatch, and layer signals reveal where adaptation is needed. Only by combining these signals with budgeted selection and driftaware updates can the model adapt without disturbing stable knowledge. Therefore, we propose DGLS to measure Dispatch-weighted Spectral Drift, update affected layers under a resource budget, and validate each candidate before deployment. DGLS should not adapt to every traffic change, because a global drift score may rise even when fleet decisions are unaffected. DGLS instead encodes a recent window C𝑡 and a time-, zone-, and demand– supply-matched reference R with the last accepted model 𝜃 ★, while retaining each sample’s graph version. For the resulting scale sets C𝑡𝑏 and R𝑡𝑏 , DGLS measures dispatch-weighted maximum mean discrepancy (MMD) and applies a hysteretic trigger: 𝑧𝑡 = Hyst(𝐷𝑡 ; 𝜏on, 𝜏off , 𝑧𝑡 −1 ),
(10)
2
Í
(C𝑡𝑏 , R𝑡𝑏 ), 𝛼 𝑏𝑡 is the gate weight of scale where 𝐷𝑡 = 𝑏 ∈ B 𝛼 𝑏𝑡 MMD 𝑏 averaged over the zones and times in C𝑡 . The operator Hyst activates adaptation when 𝐷𝑡 ≥ 𝜏on , deactivates it when 𝐷𝑡 ≤ 𝜏off , and otherwise retains 𝑧𝑡 −1 . The gate average removes dispatchirrelevant changes, and 𝜏on > 𝜏off avoids boundary-trigger noise. While 𝐷𝑡 determines when adaptation is necessary, it does not specify which layers should be updated. Updating all layers would not only exhaust the adaptation budget but could also overwrite stable knowledge. DGLS therefore identifies the layers most relevant to the current drift by jointly considering activation drift, current gradient sensitivity, and historical importance. Let 𝒜𝑙 (B; 𝜃 ) denote the activation set produced by layer 𝑙 for batch B. Using min–max normalization N𝑙 across layers, the three layer signals are ∥ ∇𝜃𝑙 Lrecent ∥ 𝐹 2 tr ★ ★ √ 𝐴𝑙 = N𝑙 MMD (𝒜𝑙 (C𝑡 ; 𝜃 ), 𝒜𝑙 (R𝑡 ; 𝜃 )) , 𝐺𝑙 = N𝑙 , |𝜃𝑙 | Í b𝑙,𝑟 ] 𝑗 . Here, Lrecent is the joint loss on the and Ω𝑙 = N𝑙 |𝜃1 | 𝑗 [ 𝛀 𝑙
b𝑙,𝑟 is the bias-corrected moving current adaptation prefix, and 𝛀 average of squared validation gradients from accepted update 𝑟 . DGLS then scores the layers and selects a positive-score subset within budget 𝐵: S𝑡 = GreedyBudget𝐵 {(𝑠𝑙 , 𝑐𝑙 )}𝑠𝑙 >0 , (11) where 𝑠𝑙 = 𝜔𝐴 𝐴𝑙 +𝜔𝐺 𝐺𝑙 −𝜔 𝐼 Ω𝑙 , 𝐴𝑙 locates changed representations, 𝐺𝑙 measures their current effect on the objective, and Ω𝑙 protects parameters that supported accepted behavior. The cost 𝑐𝑙 can represent parameters, FLOPs, or execution time; GreedyBudget𝐵 ranks positive-score layers by 𝑠𝑙 /𝑐𝑙 and adds a layer only when its cost fits Í the remaining budget, ensuring 𝑙 ∈ S𝑡 𝑐𝑙 ≤ 𝐵. All other parameters and optimizer states remain frozen.
MobiWave: Dispatch-Oriented Graph Wavelets and Drift-Guided Selective Optimization for Autonomous Fleet Rebalancing
Layer selection limits where adaptation occurs; preserving accepted behavior still requires protection against overfitting the short drift window. The selected layers therefore minimize ∑︁ Ladapt = Lrecent + 𝜆𝑟 Lref + 𝜆𝑎 ∥a𝑙 − a𝑙★ ∥ 22 𝑙 ∈ S𝑡
+ 𝜆𝐼
∑︁
b𝑙1/2 ⊙ (𝜃𝑙 − 𝜃 ★)∥ 2, ∥𝛀 2 𝑙
(12)
𝑙 ∈ S𝑡
where Lrecent fits the current drift, while Lref preserves available historical demand and return behavior. For each selected layer, a𝑙 is its mean candidate activation, a𝑙★ is the matched stable activation, b𝑙 stores parameter-level importance whose normalized layer and 𝛀 average gives Ω𝑙 in Equation (11). The last two terms therefore protect stable activations and parameters while the recent loss learns the drift. The protected objective controls what is retained; a single optimizer memory can nevertheless treat a short shock and a persistent change in the same way. For g𝑙,𝑘 = ∇𝜃𝑙 Ladapt , DGLS schedules slow writes from the shock and persistence statistics 𝜔𝑡 = 𝜔 max 𝑃𝑡sh , 𝑃𝑡 +Δ𝑡 +𝜖
where 𝑃𝑡 = 𝛽𝑃 𝑃𝑡 −1 + (1 − 𝛽𝑃 )𝐷𝑡 and Δ𝑡sh = [𝐷𝑡 − 𝐷𝑡 −1 ] + . The selected parameters are then updated by combining fast and slow optimizer memories: 𝑓
b 𝑙,𝑘 + 𝜔𝑡 m b 𝑙,𝑘 ª ©m 𝜃𝑙,𝑘+1 = 𝜃𝑙,𝑘 − 𝜂𝑙 Orth √︁ ® , 𝑙 ∈ S𝑡 , b v𝑙,𝑘 + 𝜖 ¬ « 𝑠
(13)
𝑓
where m b 𝑙,𝑘 is the bias-corrected fast momentum updated at every 𝑠 is the bias-corrected slow momentum updated only inner step, m b 𝑙,𝑘 1+𝜅 Δsh 𝑠 ,𝑇 𝑠 when 𝑘 −𝑘 − ≥ 𝑇𝑡𝑠 , 𝑇𝑡𝑠 = clip 𝑇0𝑠 1+𝜅𝑆𝑃 𝑃𝑡𝑡 ,𝑇min v𝑙,𝑘 is the max , and b bias-corrected second moment. The index 𝑘 − marks the preceding slow write, and 𝜂𝑙 is the layer-specific learning rate. A sudden increase in drift enlarges 𝑇𝑡𝑠 and suppresses 𝜔𝑡 , whereas persistent drift shortens the interval and increases the contribution of slow memory. Orth applies short Newton–Schulz orthogonalization only to matrix directions and is the identity map for vector parameters. Equation (13) therefore reacts quickly without storing a temporary spike as lasting knowledge; the complete moment recurrences are given in Appendix A. Even this protected, budgeted update remains a candidate, because limited recent data can still make it reduce dispatch reward or violate service constraints. Candidate validation therefore uses an adaptation prefix C𝑡tr and a later held-out suffix C𝑡val , both ending before decision time 𝑡. The candidate and stable models are replayed from the same fleet state under identical requests and travel times, isolating the effect of the model update. For the validation reward 𝑅val and the lower-is-better violation metric 𝑞 𝑗 , define Δ𝑅𝑡 = 𝑅val (𝜽 cand ) − 𝑅val (𝜽 ★) and Δ𝑞 𝑗,𝑡 = 𝑞 𝑗 (𝜽 cand ) − 𝑞 𝑗 (𝜽 ★). The candidate update is accepted only when Acc𝑡 = I Δ𝑅𝑡 ≥ 𝜖𝑅 ∧ Δ𝑞 𝑗,𝑡 ≤ 𝜖 𝑗 , ∀𝑗 , (14) where 𝜖𝑅 > 0 is the required validation-reward margin and 𝜖 𝑗 ≥ 0 is the allowed degradation tolerance for monitored service or safety requirement 𝑗. The positive reward margin prevents updates from being accepted due to negligible or random validation fluctuations, while
KDD ’27, August 2027, San Jose, CA, USA
the constraint tolerances reject reward-improving candidates that excessively worsen monitored service or safety requirements. Acceptance commits the candidate parameters and reference statistics. Rejection restores the stable model and optimizer state. Online deployment also requires controlled computation and memory. The shared Chebyshev basis, bounded reference buffer, layer budget, and inner-step cap bound online memory and adaptation work. Candidate validation changes the deployed state only when Equation (14) is satisfied. For completeness, Appendix A presents the end-to-end algorithm, auxiliary recurrences, and the online training process for the graph-wavelet and DGLS components.
4
Experiments
We organize the evaluation around five research questions: • RQ1: How does MobiWave compare with traditional, recent learning-based, and language-model-assisted dispatch methods? • RQ2: How reliably does MobiWave detect, adapt to, and recover from different mobility drifts? • RQ3: What is the contribution of each proposed design to dispatch quality and continual adaptation? • RQ4: What is DGLS’s updated-parameter footprint? • RQ5: How sensitive is MobiWave to its graph-scale, layer-budget, drift-trigger, and slow-memory settings?
4.1
Dataset
We use two real mobility traces and one controlled simulator. Manhattan contains 6,317 requests and 84,000 trajectory records from 350 taxis over four hours in 19 subareas, providing a compact realcity setting. Hangzhou covers 30 days, 928 subareas, 9,041 taxis, and more than 15 million requests, and therefore tests a much larger spatial and temporal scale. Simulate uses a 20 × 20 grid and timevarying Poisson arrivals. Its demand rate is the expected number of new requests per simulator step before periodic and drift multipliers are applied. Known drift onset and recovery times are hidden from policies and used only for evaluation. Table 1 reports dataset statistics for all three settings.
4.2
Experimental Settings
Baselines. Traditional baselines are DGS [7] and A-RTRS [25]. General RL baselines include TD3+BC [9], CQL [16], and Decision Transformer (DT) [5]. Dispatch-specific recent methods are NondBREM [34], GARLIC [12], CoopRide [29], and Triple-BERT [38]. We also evaluate Q policies guided by Qwen3.5:2B or Qwen3.5:9B: the language model supplies an action prior and is not a component of MobiWave. Encoder comparisons replace our first module with ChebNet [8], GWNN [30], or WaveNet [32] while retaining the same dispatch head. For online adaptation, we compare a frozen policy, full and last-layer tuning, TENT [27], LoRA [14], PALM [23], PeTTA [13], and the fixed M3 optimizer (one of the key components in [3]) inspired by multi-time-scale learning [2]. The M3 updates all layers every eight dispatch steps and fixes its slow interval and weight to 8 and 0.35, without DGLS drift scoring, budget selection, importance protection, or candidate validation.
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X. Han, P.B. Wang, Y.S. Zhu, G.J. Shen, and X.J. Kong
Table 1: Dataset statistics.
Dataset
Temporal span
Requests
Fleet / records
Spatial record
Sampling unit
Manhattan Hangzhou Simulate
4 hours 30 days 800 steps
6,317 15,144,840 Poisson arrivals
350 / 84,000 9,041 / 781,142,400 60 vehicles
19 subareas, 18 km2 928 subareas, 900 km2 20 × 20 grid
Second Minute One simulator step
Evaluation metrics. The primary fleet-level metric is the emptyloaded rate, defined as 𝑁 empty /𝑁 total × 100%, where 𝑁 empty denotes the number of active vehicle-time steps without passengers, including idle, pickup, and rebalancing steps, and 𝑁 total denotes all nonoffline vehicle-time steps. We additionally report passenger waiting time and operational profit, where profit is calculated as passenger revenue minus the costs of pickup, occupied travel, and rebalancing. Adaptation performance is evaluated using drift-detection delay, the number of false triggers, recovery steps, and forgetting on a matched historical replay. Recovery is defined as the first postdrift step at which the rolling dispatch metric returns to within 5% of its matched no-drift value, while forgetting measures the post-update reward degradation on the historical replay. Tables 2–4 mark column-best and second-best values in bold and underline, respectively; Table 5 marks only the best. Implementation details. The real traces are divided chronologically in a 6:3:1 ratio, and Simulate uses 800,000-step streams with matched requests, fleet initialization, topology, and travel times whenever a factor is not being changed. Sudden drift creates a short local demand or travel-time shock, whereas gradual drift moves the commuting distribution smoothly toward a shifted pattern. Structural drift changes road connections or vehicle travel times, recurring drift removes and later restores an event pattern, and supply-side drift temporarily reduces vehicle availability. Every drift stream has a no-drift control with the same seed and the same exogenous events outside the factor being tested. The known onset and recovery times are stored only by the evaluator, so no adaptive method receives a drift boundary. Candidate validation also uses only a causal held-out suffix whose outcomes are available before the current decision. All stochastic comparisons use ten independent runs with matched random seeds, and values are reported as means, with standard deviations shown when available. Paired bootstrap intervals and a two-sided paired permutation test at 𝑝 < 0.05 are used against the strongest baseline. Appendix B lists all hyperparameters, drift construction, and replay rules.
4.3
Overall Performance (RQ1)
We compare MobiWave with eleven traditional, learning-based, dispatch-specific, and language-model-guided policies. Table 2 reports ten-run rates across datasets varying in size, duration, and fleet scale. MobiWave ranks first by mean with 30.22% ± 2.04%, 38.54% ± 2.33%, and 29.19% ± 1.36%, respectively. Relative to the strongest competing mean, CoopRide on Manhattan and Simulate, and GARLIC on Hangzhou, the reductions are 6.58%, 5.33%, and 3.60% respectively. Causal summaries and graph-wavelet bands retain broad and local patterns, while gating favors scales that improve relocation under changing regimes.
Table 2: Empty-loaded rate on Manhattan (M), Hangzhou (H), and Simulate (S). Results are means ± standard deviations. The lower, the better. Method
Empty-loaded rate (%) ↓ M
H
S
Traditional DGS A-RTRS
32.57±1.23 32.39±1.37
41.23±2.85 41.04±2.88
30.55±0.65 30.40±1.34
Deep Learning TD3+BC CQL DT NondBREM GARLIC (GPT) CoopRide Triple-BERT Qwen3.5-2B-guided Q Qwen3.5-9B-guided Q
37.22±3.73 35.17±4.66 33.49±2.27 33.27±2.08 32.38±1.76 32.35±1.28 32.46±1.64 32.60±2.35 32.55±1.91
50.13±4.25 46.87±5.08 41.45±2.95 41.65±3.11 40.71±1.86 40.87±1.30 42.44±1.79 40.99±3.03 40.91±2.07
35.85±1.96 33.75±2.08 31.05±1.51 30.85±1.34 30.31±1.14 30.28±1.07 30.35±1.22 30.48±1.55 30.45±1.32
Ours MobiWave 30.22Mobility ±2.04 38.54 ±2.33 29.19±1.36 4.4 Adaptation under Drift (RQ2) We compare nine adaptation methods under a no-drift setting and five matched shifts in demand, traffic, topology, recurring events, and vehicle supply, as shown in Table 3. DGLS achieves the best result in the no-drift setting and across all drift types. Its average empty-loaded rate over the six settings is 28.53% ± 0.64%, compared with 29.65%±0.71% for PeTTA, corresponding to a relative improvement of 3.78%. Compared with the best competing method, DGLS improves the empty-loaded rate by 0.99 percentage points under recurring drift and by up to 2.10 percentage points under gradual drift. It also achieves 27.35% ± 0.73% under structural drift. Unlike methods based on fixed update schedules, DGLS uses dispatch-related evidence to measure spectral changes, updates only the affected layers, and combines fast and slow memory with update checks. These designs help the model adapt to mobility drift while avoiding unnecessary updates to the full model.
4.5
Ablation Study (RQ3)
We replay single-component variants over six matched streams. Table 4 reports ten-run empty-loaded rate, profit, and waiting time to cover utilization and service. Full MobiWave obtains 29.19% ± 1.36%, 79.16 ± 0.53 thousand, and 18.22±0.45 steps. Removing causal summaries raises the emptyloaded rate by 0.98 points, reduces profit by 0.67 thousand, and adds 1.07 waiting steps. Replacing graph wavelets with a GCN raises
MobiWave: Dispatch-Oriented Graph Wavelets and Drift-Guided Selective Optimization for Autonomous Fleet Rebalancing
KDD ’27, August 2027, San Jose, CA, USA
Table 3: Empty-loaded rate (%) under a matched no-drift control and five mobility-drift families. Results are means ± standard deviations. The lower, the better.
Policy Frozen policy Full tuning Last-layer tuning TENT LoRA PALM PeTTA M3 only DGLS (MobiWave)
No drift
Sudden
Gradual
Structural
Recurring
Supply side
Mean
32.33±1.37 31.78±1.24 31.28±1.33 33.59±2.03 30.65±1.42 30.45±1.33 30.22±1.36 30.38±1.32 29.19±1.36
31.30±0.72 31.12±0.5 30.92±0.87 31.98±1.25 30.98±0.67 30.67±0.38 30.72±0.19 30.68±0.28 28.66±0.50
31.27±0.92 31.03±0.67 31.52±0.82 31.96±0.86 31.44±0.85 31.02±0.89 30.55±0.91 30.96±1.23 28.45±0.79
30.31±0.61 30.11±0.62 30.08±0.72 30.55±0.69 30.46±0.74 30.04±0.77 29.31±0.82 30.04±0.71 27.35±0.73
32.44±1.08 31.64±0.88 31.31±0.86 32.56±1.17 31.51±0.82 31.24±0.76 30.54±0.79 32.05±0.93 29.55±0.69
30.85±0.87 30.88±0.65 30.82±0.72 31.18±0.94 30.79±0.66 30.36±0.62 29.77±0.58 30.44±0.91 28.02±0.71
31.42±0.84 30.79±0.65 30.65±0.75 31.65±0.87 31.20±0.64 30.51±0.89 29.65±0.71 31.04±0.83 28.53±0.64
the rate to 30.42% and waiting to 19.73, while removing dispatchaware gating raises them to 29.51% and 19.51. Causal context, scale separation, and dispatch-aware fusion are complementary. The adaptation ablations expose metric trade-offs rather than a uniform ranking. Unweighted drift increases profit to 80.31 thousand but also raises the empty-loaded rate to 30.11%; removing importance protection lowers that rate to 28.92% but increases waiting to 18.92. Thus, aggressive short-horizon updates may improve one objective by sacrificing service or stability. These safeguards restrain risky updates. The full model therefore gives the best wait and balances competing objectives.
4.6
Parameter Updates (RQ4)
Table 5 shows the number of parameters updated during adaptation under one simulation setting. DGLS updates only 62.22K parameters, compared with 135.94K for Adam, AdamW, SGD, and M3. This saves 73.72K parameters, or 54.23% of baseline updates. All four baselines update the full model, so changing the optimizer alone does not reduce the updated parameter count. DGLS instead ranks layers using dispatch-weighted drift evidence, updates a budgeted subset, and freezes the rest. This lowers gradient and optimizer-state costs while protecting unaffected layers. Thus, RQ2 gains come from focused adaptation, not model compression.
4.7
Parametric Study (RQ5)
We study the effect of four key parameters by changing one parameter at a time while keeping all other training, data-stream, and dispatch settings fixed. Figure 3 reports the ten-run results for the number of selected layers 𝐾, the layer-update budget, the drift threshold 𝜏on , and the slow-update interval 𝑇0𝑠 . As shown in Figure 3(a), the empty-loaded rate varies only from 28.26% to 28.46% when 𝐾 ranges from 2 to 6, with the best result at 𝐾 = 4. A small 𝐾 may exclude some layers related to the current drift, while a large 𝐾 may update layers that still contain useful and stable knowledge. The middle value provides enough update ability without changing too many unrelated layers. The small overall difference also shows that the layer-ranking method can consistently identify the most useful layers. A similar trend is observed for the layer-update budget in Figure 3(b), where the results remain between 28.23% and 28.39%. The
10%, 20%, and 100% budgets produce similar results because driftrelated changes are likely concentrated in a limited number of layers. Once these main layers are included, increasing the budget adds little benefit and may introduce unnecessary changes to stable layers. This explains why DGLS can achieve good performance without updating the full model. The drift threshold has a clearer effect, as shown in Figure 3(c). Setting 𝜏on = 0.05 achieves 28.18%, while thresholds of 0.2 or higher give similar results of about 28.52%. A lower threshold allows DGLS to detect changes earlier and start adaptation before the drift causes a large loss in dispatch quality. In contrast, a high threshold requires stronger evidence and may delay or skip useful updates. The similar results at high thresholds suggest that these settings lead to nearly the same late-update behavior. Finally, Figure 3(d) shows that the performance first improves and then declines as 𝑇0𝑠 increases. The best result is 28.18% at 𝑇0𝑠 = 4, compared with 28.38% at 𝑇0𝑠 = 2 and 28.52% at 𝑇0𝑠 = 32. When the slow memory is updated too often, short-term changes may be stored before their value is fully confirmed. When the interval is too long, the slow memory may retain outdated information and respond too late to lasting drift. A moderate interval therefore balances fast response and stable updates.
5 Related Work 5.1 Vehicle Dispatching and Rebalancing Vehicle dispatch has progressed from system-level guidance to learned long-horizon policies. DGS and A-RTRS combine real-time assignment with system objectives [7, 25], while CQL, TD3+BC, and Decision Transformer provide representative offline policy-learning strategies [5, 9, 16]. Recent methods constrain offline actions, coordinate city grids, or model driver–order relations [29, 34, 38]. GARLIC augments RL with multiview traffic graphs and a language-model controller [12]. These comparisons neither isolate dispatch-relevant graph frequencies nor select drift-responsive updates. Human-driven systems model relocation acceptance [4]. Autonomous fleets execute feasible platform decisions directly. This places greater weight on road representation and validated adaptation. Frozen baseline representations cannot separate harmless input changes from harmful mobility shifts.
KDD ’27, August 2027, San Jose, CA, USA
X. Han, P.B. Wang, Y.S. Zhu, G.J. Shen, and X.J. Kong
Table 4: Ten-run ablations over six mobility streams (mean ± standard deviation). Profit (103 ) ↑
Average wait ↓
29.19±1.36 30.17±0.98 30.42±0.53 29.51±1.06 30.11±1.11 29.47±0.76 28.92±0.78 29.38±1.11 29.75±0.94
79.16±0.53 78.49±0.22 80.01±0.43 80.12±0.61 80.31±0.58 79.25±0.57 79.21±0.56 79.46±0.54 79.87±0.69
18.22±0.45 19.29±0.84 19.73±0.44 19.51±0.38 18.67±0.72 18.32±0.49 18.92±0.65 19.34±0.70 19.25±0.54
Empty-loaded (%)
Empty-loaded (%)
Full MobiWave w/o causal multi-horizon summaries Graph wavelets → GCN w/o dispatch-aware gating Unweighted input drift w/o budgeted layer selection w/o historical-importance protection w/o drift-aware fast–slow update w/o candidate validation 29 28.5 28 27.5
2
3
4
5
6
(a) Graph scales 𝐾
29 28.5 28 27.5
1
5
10
20
100
(b) Layer budget (%)
29 28.5 28 27.5
0.05
0.1
0.2
(c) Trigger 𝜏on
0.4
0.8
Empty-loaded (%)
Empty-loaded rate (%) ↓
Empty-loaded (%)
Variant
29 28.5 28 27.5
2
4
8
16
32
(d) Base slow interval 𝑇0𝑠
Figure 3: Ten-run sensitivity of MobiWave (mean ± standard deviation where available). Table 5: Updated parameters per adaptation on Simulate.
Method
Updated parameters (K) ↓
Adam AdamW SGD M3 DGLS
135.94 135.94 135.94 135.94 62.22
paired candidate validation. Thus, fleet evidence governs the trigger, layer selection, and candidate deployment under one objective.
6
7 5.2
Spatiotemporal Graph Learning
Graph filters model how demand, supply, and travel conditions interact through road topology. ChebNet evaluates localized spectral filters with sparse polynomials [8], and GWNN constructs graphwavelet bases [30]. WaveNet targets nonstationary high-frequency signals [32], while WaveGC and two-dimensional filters learn richer spectral structure [6, 22]. These encoder baselines optimize representation or prediction, whereas rebalancing needs a region- and time-specific measure of which graph scale changes a move decision. MobiWave learns this measure through dispatch-aware gating and reuses it to detect future mobility drift.
5.3
Continual Adaptation
Test-time adaptation updates a deployed model without retraining on its complete history. TENT adapts normalization layers by entropy minimization [27], PALM selects layers using uncertainty and gradients [23], and PeTTA limits collapse in recurring environments [13]. FreqCTTA routes frequency shifts through adaptive experts [37], while nested learning organizes optimizer memory across time scales [2]. These baselines do not tie drift evidence and update acceptance to fleet reward and service constraints. DGLS weights spectral change by dispatch relevance, selects affected layers under a measured budget, and accepts an update only after
Ethical and Societal Considerations
MobiWave does not introduces ethical issue. It uses zone-level aggregate states and produces region-level fleet flows, not person-level decisions. These routine safeguards do not affect its contributions or conclusions.
Conclusion
We presented MobiWave for autonomous fleet rebalancing under evolving mobility conditions. It combines graph wavelets for decision-relevant frequencies with DGLS, which detects drift, updates selected layers, and rejects unsafe candidate updates. Together, the modules align representation and adaptation with fleet objectives without altering the deployed model structure. Across ten-run evaluations, MobiWave ranks first on all three datasets and DGLS ranks first under all five drift families while updating 54.23% fewer parameters than full-model optimizers. Sensitivity results favor a responsive drift trigger and an intermediate slow-memory interval. Taken together, these results show that updating fewer parameters alone does not ensure reliable online adaptation. Detected road changes must guide the choice of affected model components, and every candidate must be validated against fleet outcomes. Future work will study directed graphs, delayed demand, battery constraints, and certified fleet deployment.
MobiWave: Dispatch-Oriented Graph Wavelets and Drift-Guided Selective Optimization for Autonomous Fleet Rebalancing
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Spatio-Temporal Graph Convolutional Networks: A Deep Learning Framework for Traffic Forecasting. In Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence. 3634–3640. doi:10.24963/ijcai.2018/505 [34] Hongbo Zhang, Guang Wang, Xu Wang, Zhengyang Zhou, Chen Zhang, Zheng Dong, and Yang Wang. 2024. NondBREM: Nondeterministic Offline Reinforcement Learning for Large-Scale Order Dispatching. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 38. 401–409. doi:10.1609/aaai.v38i1.27794 [35] Rick Zhang and Marco Pavone. 2016. Control of Robotic Mobility-on-Demand Systems: A Queueing-Theoretical Perspective. The International Journal of Robotics Research 35, 1–3 (2016), 186–203. doi:10.1177/0278364915581863 [36] Sainan Zhang, Rui Mao, Jun Zhang, Luwei Xiao, and Erik Cambria. 2025. MATADOR: Multimodal traffic accident prediction enhanced by multi-source aggregated emotion recognition. Information Fusion 124 (2025), 103335. 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A
X. Han, P.B. Wang, Y.S. Zhu, G.J. Shen, and X.J. Kong
For training, the adjacency-masked probabilities define a multinomial policy over valid destinations N𝑖+ = {𝑖} ∪ { 𝑗 : (𝑣𝑖 , 𝑣 𝑗 ) ∈ E𝑡 }:
Complete Algorithm and Detailed Formulations
This appendix supplies implementation details that are not mentioned in the main methodology. It clarifies the causal feature boundary, sparse graph-filter evaluation, training targets, reference matching, optimizer memories, and candidate rollback. Algorithm 1 then connects these details into the online MobiWave procedure.
A.1
Dispatch-Oriented Multi-scale Graph Wavelet Details
A.1.1 Causal Input Construction. The main methodology defines the projected causal input Z𝑡 . Here we specify the features that are easy to implement inconsistently. Let d𝑡 , n𝑡 , n̄𝑡 = n𝑡 −s𝑡 , and b𝑡 stack demand, pre-assignment vehicles, remaining idle vehicles, and backlog over all zones. Define cyc𝑃 (𝑥) = [sin(2𝜋𝑥/𝑃), cos(2𝜋𝑥/𝑃)]. The periodic code is p𝑡 = [cyc24 (ℎ𝑡 )∥ cyc7 (𝑤𝑡 )], where ℎ𝑡 and e 𝑡 = W𝑡 + I 𝑤𝑡 are the hour-of-day and weekday indices. With W e𝑡 ] 𝑖𝑖 = Í 𝑗 [ W e 𝑡 ] 𝑖 𝑗 , the neighborhood demand–supply gap is and [ D e𝑡−1 W e 𝑡 (b𝑡 + d𝑡 − n̄𝑡 ). The arriving-vehicle feature a𝑡tr is 𝜹𝑡nbr = D obtained only from in-transit records already present in X𝑡 . Let ℎ max = max H and 𝑡 0 = 2ℎ max − 1. The stream provides X0, . . . , X𝑡0 −1 as warm-up states, and the first online decision is made at 𝑡 0 . Every temporal window used at decision time 𝑡 therefore ends at or before 𝑡, so neither the input projection nor the dispatch action uses future observations. A.1.2 Chebyshev Graph-Filter Evaluation. The heat-kernel bands are evaluated without eigendecomposition. Since the normalized Laplacian has spectrum in [0, 2], we set 𝜆¯ = 2 and e L𝑡 = 2L𝑡 /𝜆¯ − I = L𝑡 − I. For scale 𝑏 and Chebyshev order 𝑃cheb , the precomputed coefficient is
𝑐𝑏,𝑝 =
2 − 𝛿𝑝0 𝜋
∫ 𝜋 𝑔𝑏 0
¯ 𝜆 (1 + cos 𝜗) cos(𝑝𝜗) 𝑑𝜗, 2
0 ≤ 𝑝 ≤ 𝑃cheb,
(15) where 𝛿𝑝0 is the Kronecker delta. The shared responses are initialized by R0,𝑡 = Z𝑡 and R1,𝑡 = e L𝑡 Z𝑡 , and then updated by R𝑝,𝑡 = 2e L𝑡 R𝑝 −1,𝑡 − R𝑝 −2,𝑡 for 𝑝 ≥ 2. Every spectral scale reuses these responses with its own coefficients 𝑐𝑏,𝑝 . The sparse recurrence and the 𝐾 scale-specific weighted sums cost 𝑂 (𝑃cheb (|E𝑡 | + 𝐾𝑁 )𝐹𝑧 ). A L𝑡 ; the heat-scale coeffitopology change therefore rebuilds only e cients remain fixed. A.1.3 Prediction and Feasible Dispatch. In the Methodology section, we define the dispatch-aware gate and deterministic allocation rule. For reproducibility, each scale logit is produced by a onehidden-layer ReLU network 𝑓𝑔𝑏 from [H𝑏𝑖,𝑡 ∥𝜓𝑖,𝑡 ∥e𝑖,𝑡 ]. The demand head applies softplus to a linear projection of h𝑖,𝑡 , and the predicted fleet gap adds current backlog, current demand, and forecast demand before subtracting the service capacity of idle vehicles. The return head receives the endpoint representations, gap difference, travel time, and move cost; the policy head additionally receives the predicted move return.
𝜋𝜃 (U𝑡 | X ≤𝑡 , G𝑡 ) =
𝑁 Ö
Î 𝑖=1
Ö 𝑢 𝑛¯𝑖,𝑡 ! 𝑖 𝑗,𝑡 𝑝𝑖 𝑗,𝑡 . + 𝑢 ! 𝑖 𝑗,𝑡 𝑗 ∈N + 𝑖
(16)
𝑗 ∈ N𝑖
The masked softmax sets 𝑝𝑖 𝑗,𝑡 = 0 for 𝑗 ∉ N𝑖+ . During deployment, no stochastic sample is used: the model applies the deterministic largest-remainder allocation stated in the main methodology. Thus, both training and deployment preserve nonnegative integer Í flows, adjacency validity, and 𝑗 𝑢𝑖 𝑗,𝑡 = 𝑛¯𝑖,𝑡 . A.1.4 Training Targets. The main methodology gives the weighted joint objective. This subsection specifies only its supervision targets. Let V be the nonempty set of vehicles moved from 𝑖 to 𝑗, 𝑀 𝑗,𝑡 = Í𝐻 −1 𝑞 𝑣 Í𝐻 𝑞−1 Í 𝑖 𝑗,𝑡 𝑣 𝐻 𝜆𝑐 𝐶 𝑗,𝑡 +𝑞 . The 𝑖 |V𝑖 𝑗,𝑡 |, 𝑅𝑡,𝐻 = 𝑞=0 𝛾 𝑟 𝑡 +𝑞 , and 𝐶 𝑗,𝑡 = 𝑞=1 𝛾 realized per-vehicle move return is 𝜌𝑖 𝑗,𝑡 =
∑︁ 𝐶 𝐻𝑗,𝑡 1 𝑣 𝑅𝑡,𝐻 − . |V𝑖 𝑗,𝑡 | 𝑣 ∈ V max(1, 𝑀 𝑗,𝑡 )
(17)
𝑖 𝑗,𝑡
A target enters I𝜌 only after all 𝐻 outcomes and the final cancellation boundary have been observed; incomplete suffixes and empty move sets remain unlabeled. For a causal rollout batch Q, the demand loss is 𝑁
Ldemand =
𝐻
∑︁ ∑︁ ∑︁ 1 Huber𝛿𝑑 (𝑑b𝑖,𝑡 +𝑟 − 𝑑𝑖,𝑡 +𝑟 ). |Q|𝑁 𝐻 𝑖=1 𝑟 =1
(18)
𝑡∈Q
When I𝜌 ≠ ∅, the return loss is ∑︁ 1 Lreturn = 𝜌𝑖 𝑗,𝑡 − 𝜌𝑖 𝑗,𝑡 ), Huber𝛿 𝜌 (b |I𝜌 |
(19)
(𝑡,𝑖,𝑗 ) ∈ I𝜌
and it is set to zero otherwise. Let 𝐺𝑡 be the discounted rewardb𝑡 its batch-standardized value, to-go within the sampled rollout, 𝐴 and 𝜒𝑡 (𝜃 ) the ratio between the current and stored behavior-policy probabilities. The clipped policy loss is 1 ∑︁ b𝑡 , clip(𝜒𝑡 , 1 − 𝜖𝜋 , 1 + 𝜖𝜋 )𝐴 b𝑡 . (20) Lpolicy = − min 𝜒𝑡 𝐴 |Q| 𝑡∈Q
Each rollout tuple stores U𝑡 and its behavior log probability, so the denominator of 𝜒𝑡 is fixed during optimization. The average gate weight used by the balance regularizer is computed over all zones and time steps in Q.
A.2
DGLS Details
A.2.1 Drift, Reference Matching, and Layer Statistics. For each spectral scale, DGLS deterministically selects at most 256 feature rows from the recent and matched reference sets. Let 𝑘 (A, B) denote the average kernel value over all cross-set pairs. We use the Gaussian kernel 𝑘𝜐 (x, y) = exp(−∥x − y∥ 22 /(2𝜎𝜐2 )). The empirical discrepancy used by the drift score and layer diagnostics is 2 (A, B) = 𝑘 (A, A) + 𝑘 (B, B) − 2𝑘 (A, B). MMD
(21)
Before deployment, 𝜎𝜐 is fixed to the median nonzero pairwise distance of at most 512 deterministically selected reference rows, lower-bounded by 𝜖𝜎 . The kernel sums are evaluated exactly on the bounded sets in memory-bounded blocks.
MobiWave: Dispatch-Oriented Graph Wavelets and Drift-Guided Selective Optimization for Autonomous Fleet Rebalancing
The reference buffer is initialized from training history. Each recent sample is matched by time stratum, zone, and demand– supply level. If the exact stratum is empty, the closest nonempty stratum is selected lexicographically by zone-graph distance, cyclic time distance, and standardized gap difference, with disconnected zones ordered last. Recent and reference samples are encoded by the same accepted parameters 𝜃 ★ while retaining their own graph versions, which allows structural drift to be measured rather than matched away. For candidate adaptation, Lrecent is the joint loss on C𝑡tr . The reference loss contains only available demand and return labels; no policy term is computed because the compact buffer does not retain old trajectory advantages or behavior probabilities. Unlabeled reference inputs still define the stable activation anchor a𝑙★, while the candidate prefix defines a𝑙 . The main methodology defines activation drift, current gradient sensitivity, historical importance, and the budgeted layer score. The parameter-wise importance state advances only after accepted update 𝑟 : hist ⊙2 𝛀𝑙,𝑟 = 𝛽𝐼 𝛀𝑙,𝑟 −1 + (1 − 𝛽𝐼 )(g𝑙,𝑟 ) . (22) b𝑙,𝑟 = 𝛀𝑙,𝑟 /(1 − 𝛽 𝑟 ). At first For 𝑟 > 0, its bias-corrected value is 𝛀 𝐼 deployment, 𝑟 = 0 and the bias-corrected importance is defined as zero. e 𝑠𝑙 = max(0, 𝑠𝑙 )/{max𝑞 [max(0, 𝑠𝑞 )] + 𝜖} is the normalized positive score. After layers are selected under budget 𝐵, their learning rates are 𝜂𝑙 = 𝜂e 𝑠𝑙 [𝜂 min + (1 − 𝜂 min )(1 − Ω𝑙 )] , (23) where floor 𝜂 min allows an important selected layer to move cautiously instead of forcing its learning rate to zero. All hyperparameters and the positive resource budget are fixed on the adaptation split before test-time deployment. A.2.2 Fast–Slow Update. The main methodology defines the shock statistic Δ𝑡sh , persistence statistic 𝑃𝑡 , slow-write interval 𝑇𝑡𝑠 , and protected candidate objective. After the diagnostic pass, only the selected layers are trainable. Before each optimizer step, their joint gradient is clipped to global ℓ2 norm 5. Fast momentum and the second moment follow the usual exponential recurrences and are bias-corrected by the number of fast updates since the layer was activated. Let 𝑘 − denote the preÍ𝑘 ceding slow-write step and g𝑙,𝑘 = (𝑘 − 𝑘 − ) −1 𝑞=𝑘 − +1 g𝑙,𝑞 . The retained slow memory is updated only when the scheduled interval is reached: ( 𝑠 𝛽𝑠 m𝑙,𝑘 𝑘 − 𝑘 − ≥ 𝑇𝑡𝑠 , − + (1 − 𝛽𝑠 )g𝑙,𝑘 , 𝑠 (24) m𝑙,𝑘 = 𝑠 m𝑙,𝑘 − , 𝑘 − 𝑘 − < 𝑇𝑡𝑠 . The slow-write counter increments only in the first case, after which 𝑘 − is set to 𝑘; otherwise the slow state and counter remain unchanged. Newly reactivated layers reset their fast momentum, second moment, and fast counter, but retain the accepted slow memory and its write counter. With bias-corrected memories, the normalized candidate direction is 𝑓
u𝑙,𝑘 = where 𝜔𝑡 = 𝜔 max
𝑃𝑡 . 𝑃𝑡 +Δ𝑡sh +𝜖
𝑠 m b 𝑙,𝑘 + 𝜔𝑡 m b 𝑙,𝑘 , √︁ b v𝑙,𝑘 + 𝜖
(25)
KDD ’27, August 2027, San Jose, CA, USA
When the parameters are vectors, we use 𝜃𝑙,𝑘+1 = 𝜃𝑙,𝑘 − 𝜂𝑙 u𝑙,𝑘 . For a matrix parameter, reshape the direction to e u𝑙,𝑘 and transpose u𝑙,𝑘 /(∥e u𝑙,𝑘 ∥ 𝐹 + 𝜖), the it when the matrix is tall. Starting from Y0 = e short Newton–Schulz iteration is 𝜃𝑙,𝑘+1 = 𝜃𝑙,𝑘 − 𝜂𝑙 restore(Y𝑄 NS ).
(26)
where Y𝑞+1 = 23 Y𝑞 − 12 (Y𝑞 Y𝑞⊤ )Y𝑞 . Thus, a sudden change is handled mainly by fast memory, whereas persistent drift gradually contributes to the retained slow direction. A.2.3 Candidate Validation and Rollback. Candidate validation uses the held-out suffix C𝑡val , which follows the adaptation prefix but ends before the current decision time. The candidate and accepted policies are replayed from the same fleet state under identical requests, travel times, and graph evolution. If paired replay is unavailable, the candidate is rejected. Otherwise, the main-text acceptance criterion requires the validation reward to improve by at least 𝜖𝑅 and every lower-is-better service or safety metric to worsen by no more than its tolerance 𝜖 𝑗 . Acceptance atomically commits the candidate parameters and optimizer state, advances the accepted-update index, updates parameter importance with the accepted validation gradient, and appends compact validation inputs, available targets, and graph versions to the fixed-capacity reference buffer. Rejection discards the isolated candidate and leaves the accepted parameters, optimizer memories, importance state, and reference buffer unchanged.
A.3
Online Training Process
The detailed online training process is given in Algorithm 1. Its inputs are the mobility stream, the accepted model and optimizer state, the importance state and reference buffer, the hysteresis thresholds, the layer budget, and the maximum number of candidate steps, and its outputs are feasible fleet flows and the final accepted state. Line 1 initializes the drift score, persistence memory, and hysteresis state to zero before online dispatch begins. Line 2 iterates over every dispatch time from 𝑡 0 to 𝑇 . Line 3 divides observations strictly earlier than 𝑡 into a causal adaptation prefix C𝑡tr and a later held-out validation suffix C𝑡val . Line 4 uses the currently accepted parameters 𝜃 ★ and the state observed through time 𝑡 to produce a feasible fleet flow U𝑡 . Line 5 measures dispatch-weighted spectral drift against the accepted reference buffer and updates the binary trigger through hysteresis. Line 6 converts the current and preceding drift scores into the short-term shock statistic Δ𝑡sh and persistence statistic 𝑃𝑡 . Line 7 permits adaptation only when the trigger is active and both causal windows contain enough completed observations. Line 8 ranks the affected layers, selects a subset S𝑡 within budget 𝐵, and assigns their protected learning rates 𝜼𝑡 . Line 9 proceeds only when at least one layer has been selected. Line 10 copies the accepted parameters and optimizer memories into an isolated candidate state, with only the selected layers enabled for updating. Line 11 limits candidate fitting to at most 𝑀max inner optimization steps. Line 12 computes the gradient of the protected adaptation objective from the causal prefix and accepted reference samples. Line 13 applies the drift-aware fast–slow update using the gradient, shock, persistence, and layer-wise learning rates, thereby changing only the selected candidate layers and their candidate optimizer state. Line 14 compares the candidate and accepted models on the
KDD ’27, August 2027, San Jose, CA, USA
Algorithm 1: MobiWave Online Training and Adaptation Input: Stream { ( G𝑡 , X𝑡 ) }𝑡𝑇=𝑡0 ; accepted state (𝜃 ★, O★, 𝛀, R );
thresholds 𝜏on > 𝜏off ; budget 𝐵; step cap 𝑀max Output: Feasible flows {U𝑡 } and the final accepted state 1 (𝐷𝑡 0 −1 , 𝑃𝑡 0 −1 , 𝑧𝑡 0 −1 ) ← (0, 0, 0); 2 for 𝑡 ← 𝑡 0 to 𝑇 do 3 ( C𝑡tr , C𝑡val ) ← CausalSplit( G<𝑡 , X<𝑡 ); 4 U𝑡 ← Dispatch𝜃 ★ ( G𝑡 , X ≤𝑡 ); 5 𝐷𝑡 ← SpectralDrift𝜃 ★ ( C𝑡tr ∪ C𝑡val , R ); 𝑧𝑡 ← Hyst(𝐷𝑡 , 𝑧𝑡 −1 ); 6 (Δ𝑡sh , 𝑃𝑡 ) ← DriftMemory(𝐷𝑡 , 𝐷𝑡 −1 , 𝑃𝑡 −1 ); 7 if 𝑧𝑡 = 1 and Ready( C𝑡tr , C𝑡val ) then 8 ( S𝑡 , 𝜼𝑡 ) ← SelectLayers(𝜃 ★, C𝑡tr , R, 𝛀, 𝐵); 9 if S𝑡 ≠ ∅ then 10 (𝜃 cand , O cand ) ← InitCandidate(𝜃 ★, O★, S𝑡 ); 11 for 𝑘 ← 1 to 𝑀max do 12 g𝑘 ← ∇𝜃 cand Ladapt ( C𝑡tr , R ); S𝑡
(𝜃 Scand , O cand ) ←
13
𝑡
FastSlowStep(g𝑘 , Δ𝑡sh , 𝑃𝑡 , 𝜼𝑡 , O cand ); 14 15 16 17
18
Acc𝑡 ← CandidateValidate(𝜃 cand , 𝜃 ★, C𝑡val ); if Acc𝑡 = 1 then (𝜃 ★, O★, 𝛀, R ) ← Commit(𝜃 cand , O cand , C𝑡val ); U𝑡 ← Dispatch𝜃 ★ ( G𝑡 , X ≤𝑡 ); Execute U𝑡 ;
same held-out suffix under matched replay and returns the acceptance indicator Acc𝑡 . Line 15 enters the commit branch only when the candidate satisfies the reward and service constraints. Line 16 atomically accepts the candidate parameters and optimizer state, updates historical importance, and refreshes the bounded reference buffer. Line 17 recomputes U𝑡 with the newly accepted model so that the current action reflects an accepted update rather than an unvalidated candidate. Line 18 executes the resulting feasible flow; if adaptation was not ready, no layer fit the budget, or validation failed, this is the flow already produced by the unchanged accepted model in Line 4.
B Experimental Details B.1 Datasets and Preprocessing Manhattan contains 350 taxis, 84,000 trajectories, and 6,317 requests over four hours. Hangzhou contains 9,041 taxis, 781,142,400 trajectories, and 15,144,840 requests over 30 days. Both real-world datasets are divided chronologically in a 6:3:1 ratio. Simulate is a 20 × 20 grid whose demand rate 2.4 means that 2.4 new requests are expected over the whole grid in each dispatch step before timevarying spatial and temporal changes are applied. Table 1 in the main paper gives the full statistics.
B.2
Protocols and Implementation
All simulator experiments use the same default protocol: a 20 × 20 grid, 60 vehicles, an 800-step horizon, demand rate 2.4, maximum wait 15, state-hop radius 3, and ten fixed random seeds. Vehicle movement costs 0.1 per grid step, and loaded passenger travel contributes 5 per grid step to revenue. The predictor sweep covers historical, MLP, and Graph Wavelet representations with AdamW,
X. Han, P.B. Wang, Y.S. Zhu, G.J. Shen, and X.J. Kong
SGD, and the recorded DGLS configuration. The workbook labels this configuration m3. Table 4 isolates the graph-wavelet and DGLS factors with the matched MLP and AdamW replacements. Legacy runs with a different simulator protocol are excluded; a row enters the reported tables only when its manifest matches the stated configuration, seeds, and source hash. The fast-momentum, slow-momentum, and second-moment factors are (0.9, 0.99, 0.999). The slow write interval is eight steps, the maximum slow-memory weight is 0.35, and matrix updates use four Newton–Schulz iterations. The MobiWave backbone uses 20 offline pretraining epochs. All compared methods receive the same request stream, fleet initialization, action constraints, and seed. The test stream inherits the bidirectional OD counts collected in the offline history, and graph weights use feasible links and free-flow travel times; observed incident delays remain in the causal state used by the output heads. Greedy nearest and Demand balance require no training, whereas PPO and the two Qwen policies use 30 epochs. Run directories record the model identifiers qwen3.5:9b and qwen3.5:2b. The operating-condition suite changes one factor at a time: fleet size in {30, 120}, demand rate in {1.5, 3.5}, maximum wait in {5, 25}, and horizon in {600, 1000}. Qwen3.5:2B uses 20 epochs in this suite.
B.3
Drift Construction
Sudden drift introduces a localized demand or travel-time shock that represents an accident, heavy rain, or a large event. Gradual drift interpolates between historical and shifted commuting patterns over an extended interval. Structural drift closes or adds road connections or persistently changes selected edge travel times. Recurring drift removes an event pattern and later restores it. Supply-side drift changes fleet size, vehicle availability, or charging-induced downtime. Each family uses the same replay interface and has a matched no-drift control. Ground-truth onset and recovery times are retained only for evaluation.